Recursive Principal Components Analysis Using Eigenvector Matrix Perturbation

Abstract

Principal components analysis is an important and well-studied subject in statistics and signal processing. The literature has an abundance of algorithms for solving this problem, where most of these algorithms could be grouped into one of the following three approaches: adaptation based on Hebbian updates and deflation, optimization of a second-order statistical criterion (like reconstruction error or output variance), and fixed point update rules with deflation. In this paper, we take a completely different approach that avoids deflation and the optimization of a cost function using gradients. The proposed method updates the eigenvector and eigenvalue matrices simultaneously with every new sample such that the estimates approximately track their true values as would be calculated from the current sample estimate of the data covariance matrix. The performance of this algorithm is compared with that of traditional methods like Sanger's rule and APEX, as well as a structurally similar matrix perturbation-based method.

Keywords

• PCA
• recursive algorithm
• rank-one matrix update.